ENAMELS, COLOUR MATCHING MODELISATION
M. Leveaux¹, V. Duchamp¹, Céline Junius², J. Gombert², K. De Greve¹, S. Teerlinck¹, M.-F. Perrin1, D. Patou^1
1c/o PEMCO Brugge, Pathoekeweg 116, 8000 BRUGGE, Belgium
² c/o SPC SOFTWARE, 24 Rue de la Convention - Bâtiment 21, F-41300 SALBRIS., France

Introduction
As for a great number of coatings applied on a substrate, enamel also has to be coloured besides its resistance and durability qualities. In applications such as architecture, culinary articles, domestic or sanitary appliances, the visual aspect is important as well as all other more technical characteristics. As a result, the professional in the field–enamel supplier or enameller, iscontinuously confronted with new requests from designers or marketing people. Today these requests have to be satisfied in an increasingly short period.
If in the past the colour was especially a matter of practical experience, the art man having to acquire as much knowledge as possible about frits, pigments and their combinations, today we cannot content ourselves with this situation and this for flexibility, reactivity or capacity reasons.
Computer science can reinforce and complete this experience, and at the same time contribute toaccelerating the whole colour matching process. In this article, two different ways are presented.
The first one uses a preliminary studied mixture plan, which enables, by using simple mathematical laws, approaching the searched result on condition that it is possible to realise this result startingfrom this preliminary set mixture plan.
The second one examines the pigment characterisation, which in this way creates a database used by calculation software that recomposes the required colour. This system, in comparison to the previous one, has no other limit than the art limit.
The two systems that were initially used in other industries (plastics, paintings, inks) have been adapted to the enamel industry.

Use of a mixture plan
Introduction
It is often possible to present a physical or chemical property as being a complex mathematical formula of different parameters: composition, pressure, temperature, time etc. If this is not the case, it is nevertheless possible to modelise this property in the form of a purely mathematical and more or less complex function of different parameters. In this way an approached value for the examined variable is obtained and this within the limits of the model, of which the coefficients have been calculated starting from a certain number of experimental results.

Mathematical models: theoretical approach and presentation
Generalities: We consider it possible to express each of the colorimetric values characterising a colour, namely L, a and b, as being a mathematical function F of the percentages of each pigment composing this colour in the enamel mixture (Eq.1). In this way:

It has to be noted that the simpler is the mathematical function, the more the test plan to be realised will be reduced. For the same reason, the more the result could be imprecise. The larger is the test plan for a given mathematical formula, the less the result will be precise.
So, to obtain a precise mathematical modelisation, it is necessary to adapt a reduced or extremely large test plan containing numerous results, by using a complex mathematical formula.
We could consider the colour as a linear function of the composition, as Dietzel has done to estimate the thermal expansion coefficient of enamel. This mathematical model, being the easiest one, would have been used for a long time if this had been true. This is completely false when the values of L*, a* and b* are actually the reduced expression of the spectral curve characteristic of the examined colour, and which is not linear!
Quadratic Model: It examines binary interactions between each of the constituents of the mixture.
So the colour is a function of each of the percentages of the pigments and of the products of these percentages, taken two by two.
In this way the mathematical formula contains 10 terms for each of the values L*, a* and b*. So we have to realise a test plan of at least 10 trials to determine each of the functions f. This model can be used in several cases but its precision is very limited as the colour is concerned, both for enamelling and other applications (painting…). The generally observed precision is of the order 5<DE*<7. This makes that the operator can be more competitive than the model itself at this level.
Special Cubic Model: (or Reduced Cubic): It examines binary, ternary and quaternary interactions between each of the constituents of the mixture. So the colour is a function of each of the percentages of the pigments and of the products of these percentages, taken two by two, three by three, and of the product of the four (Eq. 2). In this way, we have for these four pigments:

In this way the mathematical formula contains 15 terms for each of the values L*, a* and b*. So we have to realise a test plan of at least 15 trials to determine each of the functions f. Experience has shown that you have to realise at least 19 mixtures by adding 4 validation points to obtain interesting results. This model can easily be manipulated when it requires, for at least 4 components, a reduced number of trials. It should not be used in extremely large fields, such as a mixture of 4 very different pigments (a blue, red, yellow and green one). Its precision is excellent in a reduced experimental field (e.g. pastel colours), or when at least 2 of the pigments are “neighbours”: e.g. if two of the 4 pigments are blue ones.

Cubic model: It examines binary interactions of first and second order and ternary interactions between each of the constituents (Eq. 3).
The mathematical formulas are:

Here we have 20 mathematical coefficients to be found for a test plan examining 4 pigments. So at least 20 trials have to be realised, which still remains acceptable. Practice shows that by extending the test plan to 25 mixtures, precision increases. Generally, colorimetric differences of the order 1<DE*<2.5 are obtained and it is not unusual to get results with DE*<1. This model can perfectly be used to propose new colours, when the request is not very precise. The contact person will be capable to accept directly or to refine his/her request already during the second trial.

Quartic Model: Here the mathematical formula becomes more complex (Eq. 4). For 4 pigments, we have:
L*= f_L(a, b, c, d, ab, ac, ad, bc, bd, cd, ab(a-b), ac(a-c), ad(a-d), bc(b-c), bd(b-d), cd(c-d), ab(ab)², ac(a-c)², ad(a-d)², bc(b-c)², bd(b-d)², cd(c-d)², a²bc, a²bd, a²cd, b²cd, ab²c, ab²d, ac²d, bc²d, abc², abd², acd², bcd², abcd). (4)

Identical functions for a* and b*, which only show 35 coefficients have to be determined. Experience has shown that test points are necessary. 43 trials are required to start the work. The precision is of the order DE*<1 in 100% of the cases, with a good equilibrium concerning the deviations of each of the co-ordinates L*, a* or b*.

Summary: In the following table, we have assembled the different models by giving for each of them the number of mathematical coefficients and the number of trials to be realised in function of the number of examined pigments. It is obvious that for a model and a number of examined pigments, the more extensive is the test plan at the beginning, exceeding the required test points, the more precise the result will be.

Calculations
When we suppose, in the colorimetric space resulting from the test plan that:
[C] is the matrix of the measured values L*,a* and b*
[X] is the matrix of the test plan, or better it regroups the % of each of the pigments for the realisedtrials,
[M] is the matrix of the coefficients that we are examining (Eq. 5)
So:
[C]= [X]* [M]. (5)
Two cases have to be examined.
a) The number of coefficients equals the number of trials (Eq. 6). So:
[M]= [X]^-1 [C]. (6)
b) The number of coefficients is inferior to the number of trials. So we have for the same global
formula, rectangle matrixes (Eq. 7). A diagonalisation, followed by an inversion, gives:
[M]= [X’.X]^-1 [X’].[C].
With [X’]= transposed matrix of [X]. (7)
Which comes down to applying the method of the smaller squares to the matrix calculation.

Experimental aspect
Test plan
It is obvious that the test points have to be equally spread in the whole examined space, so that the interpolation in the whole colorimetric space can be done with a good precision.
For three colouring oxides, this space of compositions is an equilateral triangle. It concerns a tetrahedron for 4 pigments. The representation for 5 pigments being a hyperspace, it can impossibly be visualised in a simple way.

Programming
Can easily be done on computer in a spreadsheet program such as Excel, which contains all the necessary functions for the analysis of the matrix system, to find all the coefficients. Once these are determined, we only need to prepare a calculation sheet for the colour research. It is possible, by an exploration of the composition space around a starting point chosen at random, to select a new point of this space corresponding to the weakest colorimetric deviation in comparison to the target. It is sufficient to use the functions of the spreadsheet program in an associated macrocommand (Fig. 2). By repeating the same operation a certain number of times, we can in this way approach the target without realising any practical trial.

Metal enamelling applications
This system enables to manage maximum 5 components. These can be 5 colouring oxides in a white enamel, to realise pastel colours. You can also make it simpler by applying it on mixtures of 3 ite frits to modify a tonality of white. You can also use 4 pigments and an opacifier, this opacifier being an additive such as titan dioxide, uverite, zircon silicate, and this in both transparent and semi-opaque enamel. But it is also possible to realise a mixture plan in a base frit (e.g. transparent), one of the variables being itself another frit (semi-opaque). In this case the number of colouring oxides is reduced to 4.
For aluminium enamels, opacification and coloration being assured by titan dioxide and colouring oxide charges, at least 4 base colours can be managed.
In case of screen pastes, five colours can be mixed without problem, on condition that they are compatible.

The same mathematical approach can perfectly be used for direct-on enamels as the colour is concerned, but you have to take into account the mixability and adherence problems to realise the mixture plan, all the mixtures not being necessarily homogeneous and/or adherent on the substrate. If it is more difficult to realise the test plan, as you have to define the limits with precision, there is no longer a continuous variation from 0 to 100%, from A in B, from A in C, from B in C etc. The calculation program enabling the colour research is also more complex. The iterations on the compositions have to take into account the space limitations of the compositions.

Example
A «quartic» mapping with 4 compounds was prepared, based on titanium dioxide, ochre, redbrown and blue stains.
The target is beige. L*=40; a*=12; b*=5
In the following table, from a starting point chosen at random, we have re-grouped the colorimetric differences obtained only by calculation on 4 iterations. We just changed the iteration on percentages as the result converges towards the target.

The only trial really prepared will be n° 5, of which we just reduced the amount of colouring oxides to assure a good coverage (lowered by 30%), this gave: L*=39.9; a*=12.31; b*= 5.12 The result corresponds to the target.

Enamel colour matching by computer assistance
Introduction
This principle is based on a preliminary gauging of each pigment compared to another one (« reference »). It allows the characterisation of the pigment in terms of colorimetric properties.
The different gaugings are integrated into a matrix, which will deduce, from a standard measurement, the ratio of each mineral pigment present into the colour measured (standard).
The required equipment is: measuring equipment (spectrophotometer; bandwidth 10 nm) and a PC with the quality control module installed. A procedure has to be set, being safe and reproducible, for the preparation of all letdowns, which are the bases for all the calculations.
This is a very important point: the final customer and the software buyer have to define together what is the modus operandi to make coloured enamels, particularly the structure of the formulation and the limit ratios for each stain. We have to avoid all kinds of problems linked to the de-mixing between different frits or bad coverage. Several questions that need to be answered before starting letdowns.

Characterisation of the pigment’s optical behaviour
Mathematical laws in effect: Pigments scatter and/or diffuse the wavelength of the incident light in a selective way. We’ll calculate its capability to absorb (K coefficient) and/or diffuse (S coefficient) the light. This phenomenon is produced from the moment that the refraction index of the object (n2) is superior to the one (n1) of the air (propagation environment).
Eyes, by nature, only see the reflected part of light (I). Detectors of spectrophotometers are placed in the scattered ray axe. Transmitted reflectance will be a calculation in percentage as below:
R = (I / I0)*100 (8)
The mathematical model was previously used for translucent applications onto a background. This was very complex. Its reduction has allowed the creation of a mathematical model for opaque applications without any influence of the background:

R∞= reflectance for an infinite thickness (total opacity)
This law is applicable to environments even weakly scattering / diffusing.
So, there is a mathematical relation between the capacity to absorb (K coefficient) and to scatter (S coefficient).
Applications of characterisation laws: To match colours, we’ll use pigments additivity properties into a mix:

By deduction, the scattering coefficient value of the mix will be:

S_mix=ΣC_iS_i               (11)

with “Ci”: concentration of each component of the mix

When we take the example of a bi-component mix [white enamel and black enamel], the additivity law will be applied:
(C₁K₁ + C₂K₂)/ (C₁S₁ + C₂S₂)= (1-R∞)² / 2R∞        (12)
With
C₁= concentration of the white paste in the mix
C₁= concentration of the black paste in the mix

In this equation, the coefficient S1 was fixed on 1. This is because it represents the perfect scattering of the white reference (scale 0-1 for S). The white paste scatters the light at each wavelength and absorbs no ray. The black paste absorbs all wavelengths of the incident light.

If, for instance C₁=100, then: 100K₁/100S₁ = (1-R∞)² / 2R∞

As we measure the sample containing 100% of the white paste, we’ll deduce the K value K₁ = (1-R₁)² / 2 * Rwhite at each wavelength.

Thus we could apply the same logic for the black enamel (but also for all coloured enamels) on each concentration in the mix [white+black] and at each wavelength We obtain a characterisation in a «multi-letdown» method (from 5 to 10 concentrations) with white enamel called «reference» (in fact, this is the one which scatters the most) and black enamel (the most absorbing) called «reference».
For each paste, a gauging curve or “smoothing curve” will be drowning at different wavelengths (400-710 nm).

K = f(C) and S = f(C)

The theory says that the capacity to absorb and/or scatter the light has to be proportional to the concentration of the pigment. Practise reveals that there is a saturation plateau of the specific concentration (the value depends on the nature of the colouring oxide). Each pigment having its own chemical nature, it will react differently in case of a high or small amount.

Experimental part & practise
Letdowns
Starting from a classic list, we will adapt it to the customer process. If he usually uses small or high concentrations, the smoothing curve will be more precise in theses specific areas.
Several criteria have to be taken into account: do formulas have fixed components? What are their natures? What kinds of components are variable? Which domains of concentration do coloured pastes cover? How is a colour made?

Measurement
Measures have to be taken as carefully as preparations were done: average of 3 measures, proper and smooth surface, correct sample presentation.

Characterization – smoothing – formula scheme used
Characterisation
Was done starting from a D/8° measurement in an opaque mode with a portable Mini-scan XE.
Enamel formulations were saved with their name – code – limit concentration and used concentrations (letdowns).

Formula scheme used
During our work at PEMCO, it was determined that a formula should always use a transparent base (we call it « BaTR”) at a constant ratio (100 %). It means that we’ll colour 100 g of transparent base with a variable percentage of coloured paste according to the colour of the standard.
This scheme will allow users to visualise colour matching results like they are used to make a colour and in accordance to physical – chemical criteria. There is no risk of making mistakes and no risk of reducing or increasing dramatically the amount of pigment in the formula.

Colour matching
Starting from the standard measurement (R %_standard) and mathematical laws, the software will do several parallel calculations to find the best composition as close as possible to the spectral curve of the standard (so, with the smallest DE*).

The standard measurement is transformed into equations:
R%standard = K_standard / S_standard for every wavelength (32).
A pigment is represented by a matrix of K coefficients (32 values) and another one of S coefficients (32 values).
If we ask the software to find a mix with 3 components, which will give the same colour as the standard, we have to respect:
K₁ + K₂ + K₃ + K_standard=1 and S₁ + S₂ + S₃ + S_standard =1 (13)
With another imperative:
(C₁K₁ + C₂K₂ + C₃K₃)/ (C₁S₁ + C₂S₂+ C₃S₃) = K_standard / S_standard(14) The software will progressively increase the concentration of the first stain to have a couple of (K,S) values near to the one of the standard (for one wavelength). The software will choose the second stain and progressively increase its concentration without changing the oncentration of the first one. This goes on for each stain.
It is the matrix calculation that allows the resolution of several equations with several unknown values.
To avoid illogical solutions and to win time, we will use the triangulation method to eliminate too divergent solutions.